Fourier transform has been a major tool in analysis for over a century. It has a serious lacking for signal analysis in which it hides in its phase information concerning the moment of emission and duration of a signal. What actually needed was a localized time frequency representation which has this information encoded in it. In 1946, Dennis Gabor [1]} filled this gap and formulated a fundamental approach to signal decomposition in terms of elementary signals. On the basis of this development, in 1952, Duffin and Schaeffer [2]} introduced frames for Hilbert spaces to study some deep problems in non-harmonic Fourier series. In fact, they abstracted the fundamental notion of Gabor for studying signal processing.
Let \(\mathcal {H}\)  be a real (or complex) separable Hilbert space with inner product \(\langle .,. \rangle .\)  A countable sequence \(\lbrace f_n\rbrace  \subset \mathcal {H}\)  is called a frame for the Hilbert space \(\mathcal {H},\)  if there exist positive constants \(A,~ B >0\)  such that
\( A\Vert  f\Vert ^2_\mathcal {H} \le \sum _{n=1}^\infty |\langle f, f_n \rangle |^2 \le B\Vert f\Vert ^2_\mathcal {H},~~~ \mbox{for all}~~ \in \mathcal {H}\) 
